MathNet 2013 Question 32

Find all continuous functions $f: \mathbb{R} \to \mathbb{R}$ having the property that $(a^2 + ab + b^2) \int_a^b f(x) \, dx = 3 \int_a^b x^2 f(x) \, dx$, for all $a, b \in \mathbb{R}$.